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Mod-p Poincar\'e Duality in p-adic Analytic Geometry

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arxiv 2111.01830 v3 pith:2CFNDOJ4 submitted 2021-11-02 math.AG math.NT

classification math.AGmath.NT
keywords dualitymathbfpoincarfiberalmostanalyticanswerscharacteristic
verification ladder T0 review T1 audit T2 compute T3 formal
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abstract

We show Poincar\'e Duality for $\mathbf{F}_p$-\'etale cohomology of a smooth proper rigid-analytic space over a non-archimedean field $K$ of mixed characteristic $(0, p)$. It positively answers the question raised by P. Scholze in [Sch13a]. We prove duality via constructing Faltings' trace map relating Poincar\'e Duality on the generic fiber to (almost) Grothendieck Duality on the mod-$p$ fiber of a formal model. We also formally deduce Poincar\'e Duality for $\mathbf{Z}/p^n\mathbf{Z}$, $\mathbf{Z}_p$, and $\mathbf{Q}_p$-coefficients.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Formal models for relative adic spaces

    math.AG 2025-07 conditional novelty 8.0 of 10

    Uniform qcqs adic spaces over any Tate affinoid base are shown to be equivalent to integrally closed formal models up to normalized formal blow-ups.

  2. Compactly supported $p$-adic pro-\'etale cohomology of analytic varieties

    math.AG 2025-01 conditional novelty 6.0 of 10

    Defines compactly supported p-adic pro-étale cohomology for partially proper rigid analytic varieties and proves a stable-range comparison with syntomic cohomology.

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